# Quantum Mechanics

Bo Gao

## Web resources for learning quantum physics

Note that both use Java. It is truly unfortunate that Java is NOT a good programming language for scientific computation. Imagine the possibilities for presenting science to the students and the general public if it were. (See my excitement and frustration when Java first came out, here.)

Let me know if you find other good resources.

## Relationship to research (using mainly my own work as examples)

This thing is obviously under construction. My plan is to gradually expand it as time allows.

Part of the motivation for doing this is my feeling of a growing gap between standard textbooks on Quantum Mechanics and its actually applications in research. While I will start out by using my own works as examples, I may expand beyond that in the future. In any case, the focus will be on things that you don't usually see in the textbooks.

### Quantum Mechanics I

#### Momentum representation

Here are some examples of using the momentum representation.

In the first example, you will find the analytic solution for an electron in a combined laser and a static magnetic field. In the second example, you will find the analytic solution for an electron in a combined laser and a static electric field. You do not have to understand other details. The goal is to get a feel for how the Schrodinger equation in the momentum space can actually be solved. Note that both belong to the special cases that we talked about in class.

#### Quantum-defect theory (QDT)

Maybe there is a way to incorporate some aspects of the quantum-defect theory into the teaching of quantum mechanics, to make the simple examples that we discuss, such as the hydrogen solutions and the harmonic oscillator solutions, immediately more useful in practical applications.

It is going to take me sometime to think about how to best do this. QDT is a theoretical framework that I care deeply about. I do not want to present it without some careful thoughts. For an earlier attempt, you can watch this video of my presentation at the Fano Memorial Symposium, July 2002 at ITAMP. You need the RealPlayer to watch it. Here are the slides that went with it. Ugo Fano

In the mean time, a simple example can be found in

where you can see how a straightforward generalization of the harmonic oscillator solution in a QDT approach allows one to understand, systematically, any potential that is asymptotically harmonic, but can differ from it in an arbitrary fashion at short distances.

Here are some of the references on QDT, from which many other references can be found.

• C. H. Greene, U. Fano, and G. Strinati, Phys. Rev. A 19, 1485 (1979).
• C. H. Greene, A. R. P. Rau, and U. Fano, Phys. Rev. A 26, 2441 (1982) .
• M. J. Seaton, Rep. Prog. Phys. 46, 167 (1983) .
• U. Fano and A. Rau, Atomic Collisions and Spectra, (Academic, Orlando, FL, 1986).
• M. Aymar, C. H. Greene, and E. Luc-Koenig, Rev. Mod. Phys. 68, 1015 (1996) .
• B. Gao, “General form of the quantum-defect theory for  −1/ra  type of potentials with a>2”, Physical Review A 78, 012702 (2008). PDF

#### Calculation of higher-order terms in perturbation theory

Many physical observables require calculations of perturbation terms higher than the first order. A simple example is the atomic polarizability, which, as discussed in class, is given by a second-order term.

For any perturbation term higher than the first order, one has the difficulty of summing over an infinite number of intermediate states. An Nth order perturbation term has N-1 such infinite summations.

For most nontrivial cases, there are basically two approaches to this problem. One is the Dalgarno-Lewis method [Proc. R. Soc. London, Ser. A 233, 70 (1955)], the other is the variational method of Gao and Starace.

The Dalgarno-Lewis method reduces the calculation of an Nth order perturbation term to the solutions of N-1 second-order inhomogeneous differential equations. The Gao-Starace method reduces it to the calculation of one or two linear-algebraic equations of the form of Ax = b, regardless of the order of the perturbation. Sample calculations up to 12th order have been carried out in the second paper quoted above.

For 2nd order perturbation, it is mostly a matter of taste as to which method to use. For higher orders, the Gao-Starace method is recommended (My opinion can obviously be biased! So try them out and come to your own conclusions).

Exercise: Calculate, using both methods, the static polarizability of a hydrogen atom in its ground state.

### Quantum Mechanics II

#### Variational Monte Carlo Method

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